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I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following,

[tex] c = \frac{(a + ib)}{\sqrt{2}} [/tex]

Now if I try to measure the intensity of this beam by a photodiode, the intensity will be proportional to-

[tex]<c^{\dag} c>[/tex]

On evaluating this, I get,

[tex]\frac{\left(<a^{\dag} a> + <b^{\dag} b> + i(<a^{\dag} b> - <b^{\dag} a>)\right)}{2} [/tex]

Now the book says, that this can be written as,

[tex]\frac{\left(<a^{\dag} a> + <b^{\dag} b> + i(<a^{\dag}><b> - <b^{\dag}><a>)\right)}{2} [/tex]

I am unable to understand this step, that is [tex]<a^{\dag} b> = <a^{\dag}><b>[/tex]

can someone please explain this.

I understand that these mode operators commute, but is this always true for any two commuting operators.

Thanks

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# Expectation value of two annihilation operators

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